We define a spherical and hyperbolic analog to the Euclidean projection body for star bodies via the gnomonic projection from the unit sphere and stereographic projection in the hyperbolid model of hyperbolic space. We then prove a spherical and hyperbolic projection inequality for these notions by using an adaption of Steiner symmetrization for spherical, respectively, hyperbolic, star bodies.

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.

Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterisation of the singularities. We use these results to prove that a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on the boundary.

We prove lower bounds on the density of regular minimal cones of dimension less than seven provided the complements of the cones are topologically nontrivial.

We develop a theory of generalized characters of local systems in $\infty $-categories, which extends classical character theory for group representations and, in particular, the induced character formula. A key aspect of our approach is that we utilize the interaction between traces and their categorifications. We apply this theory to reprove and refine various results on the composability of Becker-Gottlieb transfers, the Hochschild homology of Thom spectra, and the additivity of traces in stable $\infty $-categories.

We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants.

For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$-invariant hypersurface is given in terms of a ${\mathfrak g}$-invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$-invariant. Any ${\mathfrak g}$-invariant divisor gives rise to a ${\mathfrak g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$-equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M.

We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.

In this article, we study the behavior of complete two-sided hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Initially, we introduce the concept of the linearized curvature function $\mathcal {F}_{r,s}$ of a two-sided hypersurface, its associated modified Newton transformation $\mathcal {P}_{r,s}$ and its naturally attached differential operator $\mathcal {L}_{r,s}$. Then, we obtain two formulas for differential operator $\mathcal {L}_{r,s}$ acting on the height function of a two-sided hypersurface and, for the case where their support functions are related by a negative constant, we derive two new formulas for the Newton transformation $P_{r}$ and the modified Newton transformation $\mathcal {P}_{r,s}$ acting on a gradient of the height function. Finally, these formulas, jointly with suitable maximum principles, enable us to establish our rigidity and nonexistence results concerning complete two-sided hypersurfaces in $\mathbb H^{n+1}$.

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet })$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet })$ can be ‘corrected’ into a complex $({\mathcal {A}}_{\bullet })$ of pseudodifferential operators, where ${\mathcal {A}}_k - A_k$ is a zero-order correction within this class. The induced complex $({\mathcal {A}}_{\bullet })$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$. Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$. In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$.

The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$.

Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$.

We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha >2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.

In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement of the “complexified unit circle” $\{(z,w) \in \widehat {{\mathbb C}}^2 \colon z \cdot w = 1\}$. We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric ${}_2F_1$ functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed Möbius invariant subspaces of eigenspaces of the canonical Laplacian of $\Omega $. Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.

We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak’s $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of $\mathbb D^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where $\mathbb D:=\mathbb R, \mathbb C$ or $ \mathbb H $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $.

Let $f\;:\; M\rightarrow \mathbb{C}P^{2}$ be an isometric immersion of a compact surface in the complex projective plane $\mathbb{C}P^{2}$. In this paper, we consider the Helfrich-type functional $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)=\int _{M}(|H|^{2}+\lambda _{1}+\lambda _{2} C^{2})\textrm{d} M$, where $\lambda _{1}, \lambda _{2}\in \mathbb{R}$ with $\lambda _{1}\geqslant 0$, $H$ and $C$ are respectively the mean curvature vector and the Kähler function of $M$ in $\mathbb{C}P^{2}$. The critical surfaces of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ are called Helfrich surfaces. We compute the first variation of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ and classify the homogeneous Helfrich tori in $\mathbb{C}P^{2}$. Moreover, we study the Helfrich energy of the homogeneous tori and show the lower bound of the Helfrich energy for such tori.

This note is motivated by recent studies by Eriksson-Bique and Soultanis about the construction of charts in general metric measure spaces. We analyze their construction and provide an alternative and simpler proof of the fact that these charts exist on sets of finite Hausdorff dimension. The observation made here offers also some simplification about the study of the relation between the reference measure and the charts in the setting of $\text {RCD}$ spaces.

We show that any embedding $\mathbb {R}^d \to \mathbb {R}^{2d+2^{\gamma (d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{\gamma (d)}$ is the smallest power of $2$ satisfying $2^{\gamma (d)} \geq \rho (d)$, and $\rho (d)$ denotes the Hurwitz–Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $3$-regular maps, for infinitely many dimensions $d$, without resorting to sophisticated algebraic techniques.